Samuel J. Lomonaco

Braiding operators are universal quantum gates

This paper explores the role of unitary braiding operators in quantum computing. We show that a single specific solution R (the Bell basis change matrix)oftheYang-Baxterequationisauniversalgateforquantumcomputing,in thepresenceoflocalunitarytransformations.Weshowthatthissame Rgeneratesa new non-trivial invariant of braids, knots and links. Other solutions of theYang- Baxter equation are also shown to be universal for quantum computation. The paperdiscussestheseresultsinthecontextofcomparingquantumandtopological points of view. In particular, we discuss quantum computation of link invariants, the relationship between quantum entanglement and topological entanglement, and the structure of braiding in a topological quantum field theory.

Quantum entanglement and topological entanglement

This paper discusses relationships between topological entanglement and quantum entanglement. Specifically, we propose that it is more fundamental to view topological entanglements such as braids as entanglement operators and to associate with them unitary operators that are capable of creating quantum entanglement.

A quick glance at quantum cryptography

The recent application of the principles of quantum mechanics to cryptography has led to a remarkable new dimension in secret communication. As a result of these new developments, it is now possible to construct cryptographic communication systems which detect unauthorized eavesdropping should it occur, and which give a guarantee of no eavesdropping should it not occur.

Quantum knots and mosaics

In this paper, we give a precise and workable definition of a quantum knot system, the states of which are called quantum knots. This definition can be viewed as a blueprint for the construction of an actual physical quantum system. Moreover, this definition of a quantum knot system is intended to represent the “quantum embodiment” of a closed knotted physical piece of rope. A quantum knot, as a state of this system, represents the state of such a knotted closed piece of rope, i.e., the particular spatial configuration of the knot tied in the rope. Associated with a quantum knot system is a group of unitary transformations, called the ambient group, which represents all possible ways of moving the rope around (without cutting the rope, and without letting the rope pass through itself.) Of course, unlike a classical closed piece of rope, a quantum knot can exhibit non-classical behavior, such as quantum superposition and quantum entanglement. This raises some interesting and puzzling questions about the relation between topological and quantum entanglement. The knot type of a quantum knot is simply the orbit of the quantum knot under the action of the ambient group. We investigate quantum observables which are invariants of quantum knot type. We also study the Hamiltonians associated with the generators of the ambient group, and briefly look at the quantum tunneling of overcrossings into undercrossings. A basic building block in this paper is a mosaic system which is a formal (rewriting) system of symbol strings. We conjecture that this formal system fully captures in an axiomatic way all of the properties of tame knot theory.

Entanglement criteria: quantum and topological

This paper gives a criterion for detecting the entanglement of a quantum state, and uses it to study the relationship between topological and quantum entanglement. It is fundamental to view topological entanglements such as braids as entanglement operators and to associate to them unitary operators that are capable of creating quantum entanglement. The entanglement criterion is used to explore this connection.

Metacyclic error-correcting codes

Error-correcting codes which are ideals in group rings where the underlying group is metacyclic and non-abelian are examined. Such a groupG(M, N,R) is the extension of a finite cyclic group ℤM by a finite cyclic group ℤN and has a presentation of the form (S, T:SM=1,TN=1, T· S=SR·T) where gcd(M, R)=1, RN=1 modM, R ≠ 1. Group rings that are semi-simple, i.e., where the characteristic of the field does not divide the order of the group, are considered. In all cases, the field of the group ring is of characteristic 2, and the order ofG is odd.Algebraic analysis of the structure of the group ring yields a unique direct sum decomposition ofFG(M, N, R) to minimal two-sided ideals (central codes). In every case, such codes are found to be combinatorically equivalent to abelian codes and of minimum distance that is not particularly desirable. Certain minimal central codes decompose to a direct sum ofN minimal left ideals (left codes). This direct sum is not unique. A technique to vary the decomposition is described. p]Metacyclic codes that are one-sided ideals were found to display higher minimum distances than abelian codes of comparable length and dimension. In several cases, codes were found which have minimum distances equal to that of the best known linear block codes of the same length and dimension.

Nuclear-magnetic-resonance quantum calculations of the Jones polynomial.

The repertoire of problems theoretically solvable by a quantum computer recently expanded to include the approximate evaluation of knot invariants, specifically the Jones polynomial. The experimental implementation of this evaluation, however, involves many known experimental challenges. Here we present experimental results for a small-scale approximate evaluation of the Jones Polynomial by nuclear-magnetic resonance (NMR), in addition we show how to escape from the limitations of NMR approaches that employ pseudo pure states. Specifically, we use two spin 1/2 nuclei of natural abundance chloroform and apply a sequence of unitary transforms representing the Trefoil Knot, the Figure Eight Knot and the Borromean Rings. After measuring the state of the molecule in each case, we are able to estimate the value of the Jones Polynomial for each of the knots.

Quantum Information Science and Its Contributions to Mathematics

This volume is based on lectures delivered at the 2009 AMS Short Course on Quantum Computation and Quantum Information, held January 3-4, 2009, in Washington, D.C. Part I of this volume consists of two papers giving introductory surveys of many of the important topics in the newly emerging field of quantum computation and quantum information, i.e., quantum information science (QIS). The first paper discusses many of the fundamental concepts in QIS and ends with the curious and counter-intuitive phenomenon of entanglement concentration. The second gives an introductory survey of quantum error correction and fault tolerance, QISs first line of defence against quantum decoherence. Part II consists of four papers illustrating how QIS research is currently contributing to the development of new research directions in mathematics. The first paper illustrates how differential geometry can be a fundamental research tool for the development of compilers for quantum computers. The second paper gives a survey of many of the connections between quantum topology and quantum computation. The last two papers give an overview of the new and emerging field of quantum knot theory, an interdisciplinary research field connecting quantum computation and knot theory. These two papers illustrate surprising connections with a number of other fields of mathematics. In the appendix, an introductory survey article is also provided for those readers unfamiliar with quantum mechanics.

A Talk on Quantum Cryptography or How Alice Outwits Eve

Alice and Bob wish to communicate without the archvillainess Eve eavesdropping on their conversation. Alice, decides to take two college courses, one in cryptography, the other in quantum mechanics. During the courses, she discovers she can use what she has just learned to devise a cryptographic communication system that automatically detects whether or not Eve is up to her villainous eavesdropping. Some of the topics discussed are Heisenberg’s Uncertainty Principle, the Vernan cipher, the BB84 and B92 cryptographic protocols. The talk ends with a discussion of some of Eve’s possible eavesdropping strategies, opaque eavesdropping, translucent eavesdropping, and translucent eavesdropping with entanglement.

Spin networks and anyonic topological computing

We review the q-deformed spin network approach to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups.